Pythagorean Trigonometric Identity

A Pythagorean Trigonometric Identity is the Pythagorean theorem expressed in terms of trigonometric functions.

• AKA: Pythagorean Identity, Fundamental Pythagorean Trigonometric Identity.
• Context:
  • It can be expressed in terms of sine and cosine functions when the Pythagorean theorem is normalized to square of the hypotenuse :

[math]\displaystyle{ a^2+b^2=c^2 \Longleftrightarrow \left(\frac{a}{c}\right)^2+\left(\frac{b}{c}\right)^2=1 \Longleftrightarrow \sin^2(\theta)+\cos^2(\theta)=1 }[/math]

where c is the hypotenuse of the right triangle, a and b are the opposite and the adjacent side to the acute angle [math]\displaystyle{ \theta }[/math]

• • It can expressed in terms of other trigonometric functions when divided by the square of cosine or sine functions, respectively, as follows:

[math]\displaystyle{ 1 + \tan^2(\theta)= \sec^2(\theta)\quad\text{and}\quad 1 + \cot^2(\theta) = = \csc^2(\theta) }[/math]

where [math]\displaystyle{ \tan(\theta) }[/math] is the tangent function, [math]\displaystyle{ sec(\theta) }[/math] the secant function [math]\displaystyle{ \cot(\theta) }[/math] is the cotangent function and [math]\displaystyle{ \csc(\theta) }[/math] is the cosecant function.

• See: Cosine Function, Sine Function, Pythagorean Theorem, Tangent Function, Cotangent Function, Secant Function, Cosecant Function, Right Angle, Hypotenuse.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity
  • QUOTE: The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

The identity is given by the formula:

[math]\displaystyle{ \sin^2 \theta + \cos^2 \theta = 1.\! }[/math]

(Note that sin² θ means (sin θ)²). This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity

If the length of the hypotenuse of a right triangle is 1, then the length of either of the legs is the sine of the opposite angle and is also the cosine of the adjacent acute angle. Therefore, this trigonometric identity follows from the Pythagorean theorem.

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